A bell is a solid body that undergoes vibration and radiates energy into the air to make sound. A bell is typically a hollow body having an opening at one end (the “mouth”). In Western musical systems, bells are typically axisymmetric. Oriental bells having an oval (as opposed to a circular) cross-section are known.
It is convenient to define certain terms that are commonly used in the description of bells. A circular line on the circumference of the bell, in a plane normal to the axis of symmetry, is called a “ring”. A line on the surface of the bell, in a plane parallel to, and passing through, the axis of symmetry, is called a “meridian”. Similarly, a direction along a meridian is referred to as a “meridonal direction”.
When a bell is struck, it undergoes a complex vibration. The vibratory motion of a bell may be regarded as a linear combination of different motions of the bell known as the bell's “normal modes of vibration”, or, simply, “modes”. Each mode of vibration represents a particular vibratory shape and takes place at a single frequency of vibration. Thus, the rich sound of a bell may be regarded as being the superposition of many different frequency components, with each frequency being due to its associated mode of vibration.
The acoustically important modes are those in which displacement occurs in a direction normal to the bell's surface, as these modes are able to most efficiently radiate their energy in the air in the form of sound. In what follows, it is to be understood that a reference to “modes” is a reference only to modes in which vibration occurs in a direction normal to the bell's surface. Similarly, a reference to “frequencies” is a reference only to frequencies due to modes in which vibration occurs in a direction normal to the bell's surface.
Typically (depending on how the bell is struck), the amplitude of vibration of higher frequency modes decays more rapidly than that for lower frequency modes so that after a short time (typically of the order of one second) only several of the lowest frequency modes continue to be heard. Furthermore, complex psycho-acoustic effects are generally believed to make only the several lowest frequency modes musically important and/or discernible.
We adopt the traditional naming convention of the modes according to the number and location of nodes, or stationary lines, of the mode in question. That is, modes are referred to as an ordered pair (m,n) where m is the number of meridonal nodal lines and n is the number of nodal rings. The (2,0) mode is the lowest frequency acoustically important mode (“the fundamental”). Where there are two modes satisfying the criteria of a given ordered pair (m,n), then the lower frequency mode is referred to as (m,n) and the higher frequency mode is referred to as (m,n#).
In this specification, a reference to the first, say, three, modes is a reference to the lowest frequency mode, the second lowest frequency mode and the third lowest frequency mode. Other references to the first number of modes are to be construed similarly.
In this specification, a reference to the “mode sequence” for a bell is a reference to a list of the modes of the bell, in order of the frequency of the modes and starting with the lowest frequency mode. Similarly, references to a “frequency sequence” are references to a list of the modal frequencies of a bell starting with the lowest modal frequency.
In this specification, references to frequencies “being tuned”, and similar expressions, are references to modal frequencies which are desired to be modified to substantially adopt particular values. For example, in the case of an harmonic bell wherein the first five frequencies are to be substantially in an harmonic sequence, the first five frequencies are all “being tuned” or “to be tuned”. In a similar vein, a reference to a “tuned bell” is a reference to a bell that has modal frequencies that have been tuned.
The sound quality or timbre of a bell is dependent upon the frequency ratios and relative strengths of the different frequency components in the sound emitted by the bell. Consequently, it has long been the goal of bell founders to make bells where the frequencies of the lowest several modes are tuned as nearly as possible to particular frequency ratios. For example, church bells are typically tuned so that the first five modes have frequencies that are as near as possible to the ratios (with respect to the fundamental): 1, 2, 2.4, 3 and 4. In this instance, the first five modes have traditionally been considered to be the modes (2,0), (2,1), (3,1), (3,1#) and (4,1).
This tuning system is referred to as the “minor third” because the frequency of the third mode is musically at an interval of a minor third above the frequency of the second mode (and an interval of a minor third plus an octave from the frequency of the first mode).
Handbells are generally smaller than church bells, and generally have a different tuning. Generally, only two modes are tuned in handbells. In the so-called English tuning of handbells, the frequency of the second mode (usually the (3,0) mode) is tuned to three times the frequency of the fundamental (the (2,0) mode). Others tune the second mode to 2.4 times the frequency of the fundamental to give the bell a minor third character.
Minor third bells generally have a curved outer wall of approximately hyperbolic shape in the meridonal direction. As the wall approaches the mouth of the bell, it generally includes a relatively thick portion, prior to termination of the wall at the mouth. This thicker portion is often referred to as the “sound bow”. Handbells generally also have a thicker portion at around the same location, often referred to as the “lip”.
Due to the complexity of the vibratory motion of a bell, it is considered practically impossible to design a bell with a given frequency distribution by only physical methods and/or simple computational methods.
In 1987 a team of Dutch engineers used a numerical computation scheme to design a major third bell, where the first five modes have frequencies in the ratios (with respect to the lowest frequency): 1, 2, 2.5, 3 and 4 (“The Physics of Musical Instruments”, Fletcher and Rossing, Springer-Verlag, 1991, Chapter 21). The numerical scheme was based on the finite element method in conjunction with an optimisation algorithm. The finite element method (to be discussed further below) is, relevantly, a computational tool that is capable of numerically estimating the modes of vibration of a solid body and their associated frequencies.
Different major third bells have been designed. Each has a substantially visually different profile to that of the known minor third bell shapes. In particular, the major third bells have a curved outer surface with at least two extra turning points in the meridonal direction in comparison to the minor third bell shape.
Ideally, a bell with the greatest clarity (and, possibly, beauty) of sound would be an harmonic bell, that is a bell having at least the first several modal frequencies in the ratios 1, 2, 3, etc. Due to the complexity of bell vibrations, it has hitherto been considered impossible to design an harmonic bell. Indeed, the art of bell founding is known to date beyond 1600BC. However, an ideal or harmonic bell has never, until now, been produced. A bell with modes in an harmonic sequence would be expected to have many musical advantages over currently known bells. It is generally believed that pitch perception in humans is based on finding a best fit of an harmonic sequence to the perceived spectra of a sound, and attributing perceived pitch to the fundamental frequency of that, fitted, harmonic sequence. Bells with vibrational modes in an harmonic sequence would therefore be expected to create less ambiguous pitch perceptions.
Further, music is generally written for instruments with harmonic spectra (or very near to harmonic spectra). Consonant musical relationships are created when notes are tuned so that they have spectral frequencies that are in common and/or that are well separated. Since the spectral frequencies are whole number multiples of the fundamental frequency these notes have tended toward simple number ratios of each other in most musical tuning systems. Bells with harmonic spectra would therefore be expected to create more consonant musical relationships in most musical tuning systems and with most other musical instruments than currently known bells.
Finite element methods for numerically estimating the normal modes of a solid body, and their associated frequencies, are known. In this method, the body is notionally divided into many elements. The geometry or layout of this division is called a “mesh”. Individual elements are defined by so-called “nodes” which are points on the boundaries of elements. Many different element types, with different properties, and different areas of applicability, are known.
The finite element method, like other methods of analysing modes, is capable of estimating the mode shapes and frequencies of a body only for a particular, given, geometry.
It is known to use analysis methods, such as the finite element method, in conjunction with so-called “optimisation methods”. The goal of an optimisation method is, using certain optimisation rules, to successively modify the input to an analysis method (a given geometry in the case of analysing the modes of solid bodies) until the analysis method indicates that the performance of the input has succeeded in reaching a desired value (known as the “objective”). The performance criterion is quantified by way of the so-called “objective function”. The geometry produced by such successive modifications is said to be “optimised”. For example, in the case of optimising the shape of a bell to obtain a shape having particular modal frequencies, a given modal frequency would be the “objective function” and the optimisation method would use appropriate rules to modify the shape of the bell from a given starting shape, until the analysis method (eg the finite element method) indicated that the objective function had obtained the objective (ie that the bell had attained a shape having the desired modal frequencies).
It is known to use the finite element method in conjunction with an optimisation method to design a major third bell (as described above) and in the attempted design of a bell with “clarinet tuning” (described below).
However, a very considerable difficulty with using the finite element method together with an optimisation method to design a bell with a given frequency content is that, due to the large number of variable parameters and constraints (ie the extremely complex optimisation space), unless the starting shape is suitably close to the final optimised shape, the optimisation procedure simply will not generate a solution that meets its objective.
For example, it is known to use a cylindrical bell with a uniform wall thickness as a starting shape in an optimisation procedure in an attempt to find the shape of a bell that corresponds to “clarinet tuning”, that is a tuning in which the first four modes have the frequency ratios (with respect to the fundamental) of 1, 3, 5 and 7 (“Tuning of Bells by Numerical Shape Optimisation” by Fountain, Tomas, and Trippit). However, the best result achieved using the stated starting shape was a shape having frequencies in the ratios: 1, 2.74, 4.799 and 6.57. The errors are so large that, in practical terms, the optimisation procedure may be considered to have failed to reach its objective. Further, from a musical perspective, these errors are far too large for the bell to be useful.